$\begingroup$You should fix your notation. What is the difference between HALT(f(x),y) and HALT(f,y)? You should make sure you understand the difference between f(x) and f. And also, your question is unclear. For instance, you cannot say "HALT returns true on some y". You should presumably say something like "there exists y such that HALT(f,y) returns true".$\endgroup$
– Andrej BauerDec 25 '18 at 10:10

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$\begingroup$Is HALT supposed to be computable? Is f supposed to be partial computable?$\endgroup$
– Andrej BauerDec 25 '18 at 17:30